@DaveBenson Erdös' work is not even needed here. -- Just Dirichlet's prime number theorem (there exist primes congruent to 1 modulo $10 \cdot 2^k$ for every $k$).

@DaveBenson Ah, indeed! -- That should answer the question. -- Thank you!

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@ChristianElsholtz Sorry if I am missing something obvious -- but why does the size of the interval matter here, rather than just the average order of magnitude of the number of prime factors of $p-1$?

To avoid looking at digit counts -- wouldn't the set of primes $p$ which are congruent to 1 modulo the number of prime factors of $p-1$ work as well?

@GerryMyerson Of course it shouldn't be $o(\log(\log(\log(N))))$. I made this explicit now in the question.

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